On the Spectral and Orbital Stability of Spatially Periodic Stationary Solutions of Generalized Korteweg-de Vries Equations

2015 ◽  
pp. 285-322 ◽  
Author(s):  
Todd Kapitula ◽  
Bernard Deconinck
Keyword(s):  
Orbital Stability ◽  
Stationary Solutions ◽  
De Vries ◽  
Spatially Periodic

scholarly journals The orbital stability of the cnoidal waves of the Korteweg–de Vries equation

2010 ◽  
Vol 374 (39) ◽  
pp. 4018-4022 ◽  
Author(s):  
Bernard Deconinck ◽  
Todd Kapitula
Keyword(s):  
Vries Equation ◽  
Orbital Stability ◽  
Cnoidal Waves ◽  
De Vries

scholarly journals A Note on the Stability and Instability of Travelling Wave of Korteweg-de Vries Type: The Periodic Case

2011 ◽  
Vol 14 ◽  
pp. 57-72
Author(s):  
José R Quintero
Keyword(s):  
Orbital Stability ◽  
Type Equation ◽  
Travelling Wave ◽  
Periodic Case ◽  
Travelling Wave Solutions ◽  
Stability And Instability ◽  
Periodic Travelling Wave ◽  
De Vries ◽  
The Mean ◽  
The Stability

In this paper we adapt the work of M. Grillakis, J. Shatah, and W. Strauss, or J. Bona, P. Souganidis and W. Strauss to the periodic case in spaces having the mean zero property in order to establish the orbital stability/instability of periodic travelling wave solutions of a generalized Korteweg-de Vries type equation.

Nonlinear dynamics in horizontal film boiling

2000 ◽  
Vol 402 ◽  
pp. 163-194 ◽  
Author(s):  
CHARLES H. PANZARELLA ◽  
STEPHEN H. DAVIS ◽  
S. GEORGE BANKOFF
Keyword(s):  
Evolution Equation ◽  
Travelling Waves ◽  
Stationary Solutions ◽  
Film Boiling ◽  
Constant Thickness ◽  
Initial Development ◽  
Strongly Nonlinear ◽  
Spatially Periodic ◽  
Secondary Bifurcation ◽  
The Stability

This paper uses thin-film asymptotics to show how a thin vapour layer can support a liquid which is heated from below and cooled from above, a process known as horizontal film boiling. This approach leads to a single, strongly-nonlinear evolution equation which incorporates buoyancy, capillary and evaporative effects. The stability of the vapour layer is analysed using a variety of methods for both saturated and subcooled film boiling. In subcooled film boiling, there is a stationary solution, a constant-thickness vapour film, which is determined by a simple heat-conduction balance. This is Rayleigh–Taylor unstable because the heavier liquid is above the vapour, but the instability is completely suppressed for sufficient subcooling. A bifurcation analysis determines a supercritical branch of stable, spatially-periodic solutions when the basic state is no longer stable. Numerical branch tracing extends this into the strongly-nonlinear regime, revealing a hysteresis loop and a secondary bifurcation to a branch of travelling waves which are stable under certain conditions. There are no stationary solutions in saturated film boiling, but the initial development of vapour bubbles is determined by directly solving the time-dependent evolution equation. This yields important information about the transient heat transfer during bubble development.

Effect of ionic temperature on propagation of ion-acoustic solitary waves in a collisionless plasma of warm-ion fluid and non-isothermal electrons

1975 ◽  
Vol 14 (1) ◽  
pp. 1-6 ◽  
Author(s):  
S. G. Tagare
Keyword(s):  
Solitary Waves ◽  
Perturbation Technique ◽  
Collisionless Plasma ◽  
Stationary Solutions ◽  
Ion Acoustic Solitary Waves ◽  
Ion Acoustic ◽  
De Vries

Using a perturbation technique, we derive Modified Korteweg—de Vries (MKdV) equations for a mixture of warm-ion fluid (γ i = 3) and hot and non-isothermal electrons (γ e> 1), (i) when deviations from isothermality are finite, and (ii) when deviations from isothermality are small. We obtain stationary solutions for these equations, and compare them with the corresponding solutions for a mixture of warm-ion fluid (γ i = 3) and hot, isothermal electrons (γ i = 1).

scholarly journals The Korteweg–de Vries, Burgers and Whitham limits for a spatially periodic Boussinesq model

2018 ◽  
Vol 149 (1) ◽  
pp. 191-217 ◽  
Author(s):  
Roman Bauer ◽  
Wolf-Patrick Düll ◽  
Guido Schneider
Keyword(s):  
Original System ◽  
Bloch Wave ◽  
Energy Estimates ◽  
Periodic Medium ◽  
Wave Analysis ◽  
Amplitude Equations ◽  
Boussinesq Model ◽  
De Vries ◽  
Spatially Periodic ◽  
Bloch Wave Analysis

We are interested in the Korteweg–de Vries (KdV), Burgers and Whitham limits for a spatially periodic Boussinesq model with non-small contrast. We prove estimates of the relations between the KdV, Burgers and Whitham approximations and the true solutions of the original system that guarantee these amplitude equations make correct predictions about the dynamics of the spatially periodic Boussinesq model over their natural timescales. The proof is based on Bloch wave analysis and energy estimates and is the first justification result of the KdV, Burgers and Whitham approximations for a dispersive partial differential equation posed in a spatially periodic medium of non-small contrast.

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